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```MOTION OF A PARTICLE                       291
Case    I.   The orbit is a parabola, when v<? = —
'                            r0
2 k Case   II.   The orbit is an ellipse,   when v<?< —
TQ
2 k Case III.    The orbit is a hyperbola, when v<?> --
r°
The general expression for the velocity, which is given by equation (9), may be put in the following special forms:
2k I.    v* = — ^ when the orbit is a parabola.
(2     ]\ --- ), when the orbit is an ellipse. r     ar
(2     1\ - + -), when the orbit is a hyperbola. /     a/
The quantity a is the length of the semi-transverse axis.
222. Velocity from Infinity. — The velocity which the particle acquires in falling towards the center from a point infinitely distant from the center is called the velocity from infinity. This velocity may be computed from the energy equation. Thus
^mv2= I F dr
J<X3
r m ,
-J'f*
T
Therefore                     v*=—-
r
But the last equation is identical with the relation which gives the velocity of a particle moving in a parabolic path, therefore if a particle describes a parabolic orbit its velocity at any point of its orbit is equal to the velocity it would have acquired if it had started from infinity and arrived at that```